Free, Forced, and Damped Oscillations

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LEVELJEE Advanced

If a simple pendulum has significant amplitude (upto a factor of $1/ e$ of original) only in the period between $t = 0$ s to $t = τ$ s, then $τ$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with $b$ as the constant of proportionality, the average life time of the pendulum (assuming damping is small) in seconds is

\frac{0.693}{b}

b

\frac{1}{b}

\frac{2}{b}

A particle of mass $m$ is attached to a spring (of spring constant $k$ ) and has a natural angular frequency $ω_{0}$ . An external force $F (t)$ proportional to $cos ω t (ω \neq = ω_{0})$ is applied to the oscillator. The time displacement of the oscillator will be proportional to

\frac{m}{ω _{0}^{2} - ω ^{2}}

\frac{1}{m ( ω _{0}^{2} - ω ^{2} )}

\frac{1}{m ( ω _{0}^{2} + ω ^{2} )}

\frac{m}{ω _{0}^{2} + ω ^{2}}

LEVELJEE Advanced

A massless spring ( $k = 800 N/m$ ), attached with a mass ( $500 g$ ) is completely immersed in $1 kg$ of water. The spring is stretched by $2 cm$ and released, so that it starts vibrating. What would be the order of magnitude of the change in the temperature of water when the vibrations stop completely? (Assume that the water container and spring receive negligible heat and specific heat of mass $= 400 J/kg K$ , specific heat of water $= 4184 J/kg K$ )

1 0^{- 4} K

1 0^{- 3} K

1 0^{- 1} K

1 0^{- 5} K

Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass = 500 g, decay constant = 20 g/s, then how much time is required for the amplitude of the system to drop to half of its initial value ? (ln 2 = 0.693)

An $L$ - $C$ - $R$ circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant $b$ , the correct equivalence would be

L \leftrightarrow \frac{1}{b}, C \leftrightarrow \frac{1}{m}, R \leftrightarrow \frac{1}{k}

L \leftrightarrow m, C \leftrightarrow \frac{1}{k}, R \leftrightarrow b

L \leftrightarrow m, C \leftrightarrow k, R \leftrightarrow b

L \leftrightarrow k, C \leftrightarrow b, R \leftrightarrow m

LEVELJEE Advanced

An LCR circuit is equivalent to a damped pendulum. In an LCR circuit, the capacitor is charged to $Q_{0}$ and then connected to the $L$ and $R$ as shown below. If a student plots graphs of the square of maximum charge ( $Q_{Max}^{2}$ ) on the capacitor with time ( $t$ ) for two different values $L_{1}$ and $L_{2}$ ( $L_{1} > L_{2}$ ) of $L$ , then which of the following represents this graph correctly? (plots are schematic and not drawn to scale)